strictMono_leastExt — Mathlib

English

A simplification lemma expressing a basic membership equivalence for sets in the least extension context.

Русский

Базовая эквивалентность принадлежности для множеств в контексте наименьшего расширения.

LaTeX

$$$$Eq\\big(\\text{Set.instMembership}.mem\\, s \\;\\text{univ}\\big) = \\text{true}$$$$

Lean4

theorem strictMono_leastExt : StrictMono φ := fun i j h ↦
  by
  have least := isLeast_leastExt (F := F) (E := E)
  by_contra!
  obtain eq | lt := this.eq_or_lt
  · exact (least j).1 (subset_adjoin _ _ ⟨i, h, congr_arg b eq.symm⟩)
  · refine ((least i).2 <| mt (adjoin.mono _ _ _ (image_mono ?_) ·) (least j).1).not_gt lt
    exact fun k (hk : k < i) ↦ hk.trans h

Похожие утверждения